The 1-2-3 Conjecture for Hypergraphs

A weighting of the edges of a hypergraph is called vertex-coloring if the weighted degrees of the vertices yield a proper coloring of the graph, i.e., every edge contains at least two vertices with different weighted degrees. In this paper we show that such a weighting is possible from the weight set {1,2,...,r+1} for all hypergraphs with maximum edge size r>3 and not containing edges solely consisting of identical vertices. The number r+1 is best possible for this statement. Further, the weight set {1,2,3,4,5} is sufficient for all hypergraphs with maximum edge size 3, up to some trivial exceptions.Comment: 12 page

The computational complexity of approximation of partition functions

This thesis studies the computational complexity of approximately evaluating partition functions. For various classes of partition functions, we investigate whether there is an FPRAS: a fully polynomial randomised approximation scheme. In many of these settings we also study “expressibility”, a simple notion of defining a constraint by combining other constraints, and we show that the results cannot be extended by expressibility reductions alone. The main contributions are: -� We show that there is no FPRAS for evaluating the partition function of the hard-core gas model on planar graphs at fugacity 312, unless RP = NP. -� We generalise an argument of Jerrum and Sinclair to give FPRASes for a large class of degree-two Boolean #CSPs. -� We initiate the classification of degree-two Boolean #CSPs where the constraint language consists of a single arity 3 relation. -� We show that the complexity of approximately counting downsets in directed acyclic graphs is not affected by restricting to graphs of maximum degree three. -� We classify the complexity of degree-two #CSPs with Boolean relations and weights on variables. -� We classify the complexity of the problem #CSP(F) for arbitrary finite domains when enough non-negative-valued arity 1 functions are in the constraint language. -� We show that not all log-supermodular functions can be expressed by binary logsupermodular functions in the context of #CSPs

A Note on Edge Weightings Inducing Proper Vertex Colorings

Political variables play a crucial role in shaping tax systems and tax reforms in Europe. This paper presents new evidence on the relationship between political factors and tax design in European countries. For the period 1995-2012, we find that, when the country’s ruling coalition is left-wing, income taxes become more relevant. Moreover, when the largest government party is more powerful, the share of property taxes over GDP increases, while the number of seats held by the second government party is positively associated with the level of consumption taxes over GDP. We also find that a more powerful government is associated with a lower top tax rate of the personal income tax (PIT). Building upon this evidence, we review the main arguments developed by the empirical political economy literature on taxation and discuss the role of political factors in fundamental issues of taxation: redistribution and progressivity; tax design and its impact on growth; tax complexity; tax administration; tax competition; and the need for tax reforms